-
In this and the next two exercises, we find some approximate
eigenvectors, in the sense of Corollary 8.1.29 (page 493).
For example, let the 3x3 matrix B be defined as in the 9-th
homework set, and let the sequence G(n) be as defined there,
with the starting values G(0) = G(1) = G(2) = 1. For a given m,
say m = 10, let x be the the transpose of (G(m),G(m+1),G(m+2)).
what are the best values of alpha and beta satisfying the
hypothesis of 8.1.29, namely
alpha x <= B x <= beta x ?
(By best, one means the greatest alpha and the smallest beta.)
Hence what estimate does 8.1.29 provide for rho(B) ?
Let Dn be the n x n matrix
0 1 0 0 ... 0 0 0
1 0 1 0 ... 0 0 0
0 1 0 1 ... 0 0 0
...
0 0 0 0 ... 1 0 1
0 0 0 0 ... 0 1 0
Using the transpose of (5,8,10,10, ... ,10,10,8,5)) as an
approximate eigenvector for Dn, show that rho(Dn) lies
between 1.6 and 2.0 (for n > 3).
Optional. Verify that the transpose of
(sin pi/(n+1), sin 2pi/(n+1), ... , sin n pi/(n+1))
is a positive eigenvector of Dn. What is its eigenvalue? Hence,
what is the precise value of rho(Dn)? Then verify that, as n
goes to infinity, the limit of rho(Dn) is 2.0. (You may have
to brush up your trigonometry!)
Let C be the 3x3 matrix
1 1 1
1 2 1
1 1 1
and let x be the transpose of (12,17,12). What are the best
alpha and beta for which one may invoke 8.1.29? What bounds does
one therefore obtain on the top eigenvalue rho(C)?
Next, find another approximate eigenvector, with integer entries,
which yields a sharper estimate of the eigenvalue rho(C).
Finally, find the precise value of rho(C), and an associated
eigenvector. (If you can _guess_ an eigenvector, from what has
come before, so much the better!)
This exercise may help elucidate the power method, which was
introduced in the somewhat mysterious Exercise 7 on page 63,
which we had for February 16. Here we are able to carry out
the power method for C, and moreover analyze the situation
with the methods of Section 8.2.
Let us continue with the matrix C of the previous exercise.
According to Theorem 8.2.8 (page 499), the powers of C/rho(C)
approach a very simple matrix L. All columns of L are multiples
of a single vector x, where x is an eigenvector with eigenvalue
rho(C). Moreover, according to clause (j) on page 498, the
convergence with respect to the infinity-norm is of the order
of K r^n, where r is any number larger than the quotient of
the next-to-top eigenvalue to the top eigenvalue, and K is a
constant.
What is this ratio for the matrix C? What is the eighth power
of this ratio? Compute the eighth power of C. (Don't worry about
the scalar factor that should really be there in a fastidious
invocation of Theorem 8.2.9.) Let c be the first column of the
eighth power of C. How close is c to being an eigenvector of C?
(You could answer this question by again obtaining best alpha and
beta for an application of 8.1.29. One expects alpha and beta
to agree within roughly alpha*r^8.)
Optional. Carry out a similar power analysis of the matrix B from
the 9-th Exercise Set, including the rate of convergence. You should
already have found (above) an approximate value of rho(B), but for
this exercise will also require an approximate value of the next-to-
largest eigenvalue. This may be obtained either by deflation (Ex 8,
page 63), or simply by dividing the characteristic polynomial by
(x - r), where r is the approximate value of rho(B).